Mathematics for Modeling and Scientific Computing

Thierry GOUDON, Senior research scientist INRIA, Sophia Antipolis Mediterranee Research Centre & Labo. J. A. Dieudonne, University of Nice Sophia Antipolis & CNRS, France

• Preface ixChapter 1. Ordinary Differential Equations 11.1. Introduction to the theory of ordinary differential equations  11.1.1. Existence–uniqueness of first-order ordinary differential equations 11.1.2. The concept of maximal solution  111.1.3. Linear systems with constant coefficients  161.1.4. Higher-order differential equations 201.1.5. Inverse function theorem and implicit function theorem  211.2. Numerical simulation of ordinary differential equations, Euler schemes, notions of convergence, consistence and stability  271.2.1. Introduction  271.2.2. Fundamental notions for the analysis of numerical ODE methods 291.2.3. Analysis of explicit and implicit Euler schemes  331.2.4. Higher-order schemes 501.2.5. Leslie’s equation (Perron–Frobenius theorem, power method)  511.2.6. Modeling red blood cell agglomeration 781.2.7. SEI model 871.2.8. A chemotaxis problem  931.3. Hamiltonian problems 1021.3.1. The pendulum problem  1061.3.2. Symplectic matrices; symplectic schemes 1121.3.3. Kepler problem  1251.3.4. Numerical results 129Chapter 2. Numerical Simulation of Stationary Partial Differential Equations: Elliptic Problems  1412.1. Introduction  1412.1.1. The 1D model problem; elements of modeling and analysis  1442.1.2. A radiative transfer problem 1552.1.3. Analysis elements for multidimensional problems 1632.2. Finite difference approximations to elliptic equations 1662.2.1. Finite difference discretization principles  1662.2.2. Analysis of the discrete problem 1732.3. Finite volume approximation of elliptic equations 1802.3.1. Discretization principles for finite volumes 1802.3.2. Discontinuous coefficients  1872.3.3. Multidimensional problems 1892.4. Finite element approximations of elliptic equations  1912.4.1. P1 approximation in one dimension 1912.4.2. P2 approximations in one dimension  1972.4.3. Finite element methods, extension to higher dimensions  2002.5. Numerical comparison of FD, FV and FE methods  2042.6. Spectral methods  2052.7. Poisson–Boltzmann equation; minimization of a convex function, gradient descent algorithm 2172.8. Neumann conditions: the optimization perspective  2242.9. Charge distribution on a cord 2282.10. Stokes problem  235Chapter 3. Numerical Simulations of Partial Differential Equations: Time-dependent Problems  2673.1. Diffusion equations  2673.1.1. L2 stability (von Neumann analysis) and L∞ stability: convergence  2693.1.2. Implicit schemes  2763.1.3. Finite element discretization 2813.1.4. Numerical illustrations  2833.2. From transport equations towards conservation laws  2913.2.1. Introduction  2913.2.2. Transport equation: method of characteristics 2953.2.3. Upwinding principles: upwind scheme 2993.2.4. Linear transport at constant speed; analysis of FD and FV schemes  3013.2.5. Two-dimensional simulations  3263.2.6. The dynamics of prion proliferation 3293.3. Wave equation 3453.4. Nonlinear problems: conservation laws 3543.4.1. Scalar conservation laws 3543.4.2. Systems of conservation laws  3873.4.3. Kinetic schemes  393Appendices  407Appendix 1  409Appendix 2  417Appendix 3  427Appendix 4  433Appendix 5  443Bibliography 447Index  455